Mine are still greater

As many of you know, I am currently on pre-tenure leave. I spent the Fall in Paris doing research at Jussieu (thanks in large part to the AWM mentoring travel grant), and I’m currently stationed at the University of Texas at Austin, where I am continuing my Paris project and working on a couple of others. It has definitely been a different experience for me to work exclusively on research and to have no teaching responsibilities. Even in graduate school, I taught a class or graded every single semester. I am definitely loving the chance to dedicate myself to doing research, but this is inevitably accompanied by a not-so-fun feeling of inadequacy and ignorance. I have been thinking about this feeling a lot lately, and I thought I would share some of my thoughts on this post (even though this will not be new to many of you).

In the Fall, for example, I spent about a month trying to prove something that had essentially already been proven by someone else. The proof was not written down or published anywhere, but the mathematician who had proved it had given me a sketch of how he had done it. I had proved some of the steps, but was stuck on the last one. One day, I decided to try something different (not using his sketch). It took me that one day to finish it. Now, if I had been a bit more confident in my skills, maybe I could have saved that month’s work. On the other hand, maybe it was thinking about this problem for so long that made the last minute change so simple. Maybe I just needed to get used to these ideas before I could figure it out. So sometimes I do believe the struggle, although bad for one’s self-esteem in the short term, really pays off. In fact, I believe that in large part our success as mathematicians has more to do with our ability to keep working on a problem even after failing numerous times than with our “intelligence”.

I once told a friend that working on research always makes me feel stupid, and she laughed at me. She, a humanist, couldn’t understand how I, a mathematician, could possibly think I was bad at math. Obviously, I’m not “bad at math” in the traditional sense (although I don’t believe anyone is, but that is part of a different story). But the thing is, the more math you know, the more difficult interesting problems are. As a matter of fact, I think that the feeling of inadequacy is always there, because you’re trying to solve a difficult problem. The difference is that what is difficult to me is not the same as what is difficult to a second grade math student, but I can still understand how the second-grader feels. I think Einstein perfectly captured this idea when he said: “Do not worry about your difficulties in mathematics. I can assure you, mine are still greater.” So even Einstein, one of the most popular representatives of “genius”, felt inadequate doing mathematics.

Of course, like I said earlier, being good at mathematics has more to do with perseverance than genius (at least, in my opinion). This actually leads me to a not-at-all-famous quote by a friend’s Ph.D. advisor: “To be good at math research, it’s more important to really love it and to work really hard than to be smart. You really love math and work really hard.” I don’t think she realized the backhandedness of her compliment, and truly meant it as a regular compliment (my friend, however, keeps bringing this up as both the nicest and meanest thing anyone has ever said to him). But I believe this to be true, and in fact this is the only thing that keeps me working hard at math, even though many times I feel like an idiot. First of all, I’m not really an idiot (although I don’t think I would call anyone that… well, OK, if they made me really angry maybe). But most importantly, my success does not depend so much on my intelligence (or any other innate feature that is hard to change), but on my ability to keep working on and learning about the problem until I find a solution (and sometimes even after… many times I’ve found that my proof is actually wrong or needs some fixing).

So today, instead of struggling with the computation I’ve been trying to do for the past few weeks, I have decided to write about how I feel when I struggle with a computation. I guess it’s not the most productive use of my frustration, but I do feel a little better after writing this down. I guess now I should get back to it.

So, dear readers, how do you feel about your difficulties with mathematics (assuming you have any)?

12 Responses to Mine are still greater

This is a great post, and very timely for me. I’ve been working on a few research projects, for the last month or two, which just aren’t working out at all (my heavy teaching load this semester isn’t helping). Yesterday I started asking myself (as I very occasionally do) what I’m even doing in math, if I can’t make progress on anything, and whether or not I should just find a new career. And then today I got a referee report from a good journal, filled with positive comments, for a paper that I submitted last fall, and I remembered that around a year ago I was having exactly these feelings about that project. I think it’s great to remind everyone (especially more junior mathematicians) that a lot of us have these feelings of stupidity and ineptitude from time to time, and that it’s almost always temporary. It’s easy to think that other people are smarter than you, and don’t have these doubts, because you often only see their final output (damn you, daily arXiv e-mail!).

Patrick, you have it so right. Feelings of stupidity and ineptitude must be no more or less than the exact right kind of feelings for all of us when we’re confronted with the staggering mountain range of mathematics reared up over thousands of years of human life. We want wings, but all we get, for the most part, is a climber’s axe and a rope.

And we do fall. Oh how we can fall. The one painful and glorious thing about it is that we get up again, gather ourselves, and find a new path for ourselves up the rock face to where the sun shines. That is doing mathematics.

I was always more productive in semesters when I had a light teaching load (one class) compared with no teaching. Why? I think I get really beaten down by research if that’s all I’m doing. My confidence gets shaken. At some point I say, “screw it… I can’t do this! I’m going to watch some bad TV.”

But teaching… I can do that! I can do the math. Correctly. Every time. I can answer their questions. And this math that seems hard to them, it’s easy for me. Plus, I’m actually a pretty good teacher. My students like me and respond to me. They learn stuff. Some of them even get enthusiastic about it.

So for me, teaching is a confidence boost. It strokes my mathematical ego enough that I can dive back into the research feeling more confident. Even if I’m still stuck… remember, calculus used to be hard? Right? And now this is hard. But it won’t always be.

But… and this is key… the teaching one class part is really optimal for me. If I teach more than that, I have a harder time just finding the time to get so immersed in the research that the problems become part of my subconscious.

I’m totally with Michelle on this one. I spent a summer without teaching a couple of years ago, on the premise that “I really need to get some s*** done”. I spent the entire summer (and part of the previous spring) trying to do something and failing. Over and over and over and over. Finally, a week or so before school started back up, we decided to abandon what I was trying to compute and do something else. We never ended up going back to that computation because we got around it a different way that never would have occurred to us without moving on. I sometimes wonder what would have happened if I had abandoned that computation much earlier…

Now, I’m teaching two sections of the same course this semester (because my school is awesome to new faculty), and I still can’t manage to get anything done research-wise. Argh! Any suggestions on how to step away from the teaching responsibilities in a reasonable way? I mean, without saying “No homework for the rest of the semester!” and keeping my classroom at least occasionally active?

This is very hard for me to write, because I am not the kind of mathematician who writes and publishes papers, goes to conferences, or feels sufficiently confident to engage in proof debates and struggles with others. I have a cherished M.A. in mathematics which I acquired in 2006 when I was 64 years old, after a lifetime of practical application of lower-level computational skills and learning, and several failed efforts at continuing mathematics studies that had aborted in 1961.

Mathematics is everywhere for me, and my focus is on studying and developing cosmological models, with a great fascination for those that rely on fractal dimension. I may yet write the papers I fear so much to develop; maybe I will live long enough to do that.

I agree with Dr. Salerno about two major things: the sense of inadequacy and ignorance one feels, and the deep need to persist in wrangling problems until the path through them emerges clearly. Once I wrote in a discussion thread that physics is something that cannot be done alone, but it is also something that must be done alone. I think the same thing is true of mathematics; we are caught between the need to find assurance, validation, and redirection and the need to explore for ourselves this appallingly jagged and unforgiving mental terrain.

I am grateful to you, Dr. Salerno, for your piece, and I wish you long and distinguished success by whatever terms you choose to measure it.

Hello. i just wrote to say that your “story” inspired me.
though i, a 21 yr old always had a greater inclination in maths than anything else,i didn’t pursue it as much as i should have( i now believe).also the fact that i couldn’t even clear one level of maths olympiad didn’t do anything to cheer.and in college where we had some compulsory courses for first 2 years, certain concepts took time(to the tune of 3 months) to be cleared.
anyways, i am a earth science major now.but i feel a sense of inadequancy and deep regret for not having been able to make myself proficient in it (how far could i have gone myself,no idea still). but after hearing your persistence, i am certainly enthused to take up maths after my studies are over, even if not possible in a professional sense.i simply hope to make it enough to be able to understand most nuances of modern mathematical reasearch.and of course challenge myself with all kinds of problems 🙂

Paprika, I admire your willingness to work to overcome the obstacles you face. If you’d like some encouragement and support, and some suggestions about ways to engage in newer ways with mathematics, look me up on Facebook under my name. I’m also on LinkedIn.
Warmly,
Dana

Thank you for this post. I’m an undergrad and teacher to my homeschooled chidren (my class lives in my house), and, like you, am constantly reminding myself that math is an uphill battle. One of my favorite frequent moments, though, is when the three of us are working on our math, and they see me struggle. They think I’m reasonably good at math, so when they struggle with their own work they’re less inclined to conclude that they’re stupid, since they see their mentor struggle, persevere, and eventually succeed (most of the time).

Is teaching 15 hours per week a big teaching load? Because, once I put that aside, I could work intensely in some problems and finally make some progress. And, yes, not figuring out quickly what others have done (and how can you profit from it) is rather discouraging.

I loved this article. As someone who has not been able to do “real” maths for years because I needed to work in IT in order to pay the bills I felt really inadequate and over the top when doing even stuff that should have been simple like the problems on the Project Euler website or just remembering proofs I could rederive easily a decade or so back.

All this led me to try to shift my career to “Data Science” where I would have to rebuild my knowledge of maths.

So thanks for an article that tells me I have not got stupid over the years.

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